Interpretation of interaction terms between continuous and binary factors in INLA (only one level appearing)

[Xiaoyue Liu]

Dear all,

I am fitting a Poisson regression model in INLA that includes an interaction between a continuous variable and a binary factor (sex, with levels “men” and “women”). My goal is to estimate sex-specific slopes for each continuous variable.

My model formula is roughly as follows (simplified for clarity):

y ~ f(strata, model = "iid", hyper = list(prec = list(initial = log(1e-04), fixed = TRUE))) +
x1 + x2 + ... + xN +
x1:sex + x2:sex + ... + xN:sex - 1
where sex is a factor with levels "men" and "women".

In the model summary, I obtain coefficients for:

• each main effect (x1, x2, …),
• interaction terms like x1:sexwomen, x2:sexwomen, …,
• and only one term for x1:sexmen (not for all others).

I’d like to confirm my interpretation and understand the reason for this asymmetric coding:

• For the reference group (men), the effect should correspond to the main slope (e.g., x1).
• For the other group (women), the slope should be the sum of the main and interaction effects (x1 + x1:sexwomen).

Is this interpretation correct in the context of INLA’s design matrix?

And why might one interaction column (e.g., x1:sexmen) appear in the output, while the others are dropped? Could this be related to the -1 in the formula or INLA’s internal factor-contrast handling?

Any clarification or reference on how INLA constructs these factor-interaction design matrices would be very helpful.

Thank you in advance for your insights!

Best,

Xiaoyue

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[Helpdesk (Haavard Rue)]

The parsing of the fixed effects is done as with `lm()': model.matrix() (or its
sparse-matrix version of it)

You can check this like this:

> dat=data.frame(x=1:3, z=as.factor(c("a","b","c")),y = rep(NA,3))
> inla(y~x*z, data=dat)$model.matrix
"dsparseModelMatrix": 3 x 6 sparse Matrix of class "dgCMatrix"
(Intercept) x zb zc x:zb x:zc
1 1 1 . . . .
2 1 2 1 . 2 .
3 1 3 . 1 . 3
@ assign: 0 1 2 2 3 3
@ contrasts:
$z
[1] "contr.treatment"

> inla(y~x*z, data=dat, control.fixed=list(expand.factor.strategy =
"inla"))$model.matrix
"dsparseModelMatrix": 3 x 8 sparse Matrix of class "dgCMatrix"
(Intercept) x za zb zc x:za x:zb x:zc
1 1 1 1 . . 1 . .
2 1 2 . 1 . . 2 .
3 1 3 . . 1 . . 3
@ assign: 0 1 2 2 2 3 3 3
@ contrasts:
named list()


see ?control.fixed

note that the "inla" expansion does not drop factors, so you might need to use
some kind of sum-to-zero constr to make them likehood identifiable

On Wed, 2025-10-15 at 17:32 -0700, Xiaoyue Liu wrote:
> Dear all,
>
> I am fitting a Poisson regression model in INLA that includes an interaction
> between a continuous variable and a binary factor (sex, with levels “men” and
> “women”). My goal is to estimate sex-specific slopes for each continuous
> variable.
>
> My model formula is roughly as follows (simplified for clarity):
>
> y ~ f(strata, model = "iid", hyper = list(prec = list(initial = log(1e-04),
> fixed = TRUE))) +
>      x1 + x2 + ... + xN +
>      x1:sex + x2:sex + ... + xN:sex - 1
> where sex is a factor with levels "men" and "women".
>
> In the model summary, I obtain coefficients for:

>  * each main effect (x1, x2, …),
>  * interaction terms like x1:sexwomen, x2:sexwomen, …,
>  * and only one term for x1:sexmen (not for all others).

> I’d like to confirm my interpretation and understand the reason for this
> asymmetric coding:

>  * For the reference group (men), the effect should correspond to the main
> slope (e.g., x1).
>  * For the other group (women), the slope should be the sum of the main and

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> .

--
Håvard Rue
he...@r-inla.org